4.3 Accurate Computational Models
131
a list of all five possible excited determinants
ψ
(1) = c 1 |abp|+c 2 |abp|+c 3 |aap|+c 4 |bbp|+c 5 |abp|
(4.53)
where the hh-part has been omitted for simplicity. The contracted wave function is
shorter:
ψ
(1) = c 1 {λ|abp|+µ|bbp|} + c 2 λ|abp|+c 3 {λ|aap|+µ|abp|} + c 4 µ|abp| (4.54)
Since the determinants of the second and fourth term are the same, the wave function
presents a linear dependence, which should be removed and further reduces the
number of coefficients.
The contraction written in Eq. 4.50 is used in CASPT2 and in the partiallycontracted variant of NEVPT2. The latter method is also available in a stronglycontracted variant of NEVPT2, in which the contracted external functions are
grouped together depending on the number of electrons added or removed from the
active space. In this way a reduced set of orthogonal external functions is generated.
4.3.4 Spin Unrestricted Methods
The observation that except for the state of maximum multiplicity, spin states cannot
be rigorously represented with a single determinant makes it very interesting to look
at the possibility to study magnetic interactions in a spin unrestricted setting using
a single determinant description of the spin states. We will start with the spatially
symmetric 2-electron/2-orbital case and afterwards generalize for systems with more
unpaired electrons.
The most widely applied approximation to extract J within a single determinant
description of the spin states is the so-called Broken Symmetry approach which uses
two determinants:
Φ BS =|φ 1 φ 2 |
Φ HS =|φ 1 φ 2 |
(4.55)
where the closed-shell orbitals have been omitted for convenience. The ˆ
S 2 expectation
value of Φ HS is not exactly equal to 2 since the closed shell spin orbitals appear in
pairs with slightly different spatial orbitals. However in most cases it is close to 2
and is generally considered as a good approximation to the triplet state obtained in
a spin-restricted setting.
Φ HS ≈ Φ T
(4.56)
E HS ==Φ HS | ˆ
H|Φ HS ≈E T
(4.57)
ˆ
S
2 HS ==Φ HS | ˆ
S
2 |Φ HS ≈2
(4.58)
131
a list of all five possible excited determinants
ψ
(1) = c 1 |abp|+c 2 |abp|+c 3 |aap|+c 4 |bbp|+c 5 |abp|
(4.53)
where the hh-part has been omitted for simplicity. The contracted wave function is
shorter:
ψ
(1) = c 1 {λ|abp|+µ|bbp|} + c 2 λ|abp|+c 3 {λ|aap|+µ|abp|} + c 4 µ|abp| (4.54)
Since the determinants of the second and fourth term are the same, the wave function
presents a linear dependence, which should be removed and further reduces the
number of coefficients.
The contraction written in Eq. 4.50 is used in CASPT2 and in the partiallycontracted variant of NEVPT2. The latter method is also available in a stronglycontracted variant of NEVPT2, in which the contracted external functions are
grouped together depending on the number of electrons added or removed from the
active space. In this way a reduced set of orthogonal external functions is generated.
4.3.4 Spin Unrestricted Methods
The observation that except for the state of maximum multiplicity, spin states cannot
be rigorously represented with a single determinant makes it very interesting to look
at the possibility to study magnetic interactions in a spin unrestricted setting using
a single determinant description of the spin states. We will start with the spatially
symmetric 2-electron/2-orbital case and afterwards generalize for systems with more
unpaired electrons.
The most widely applied approximation to extract J within a single determinant
description of the spin states is the so-called Broken Symmetry approach which uses
two determinants:
Φ BS =|φ 1 φ 2 |
Φ HS =|φ 1 φ 2 |
(4.55)
where the closed-shell orbitals have been omitted for convenience. The ˆ
S 2 expectation
value of Φ HS is not exactly equal to 2 since the closed shell spin orbitals appear in
pairs with slightly different spatial orbitals. However in most cases it is close to 2
and is generally considered as a good approximation to the triplet state obtained in
a spin-restricted setting.
Φ HS ≈ Φ T
(4.56)
E HS ==Φ HS | ˆ
H|Φ HS ≈E T
(4.57)
ˆ
S
2 HS ==Φ HS | ˆ
S
2 |Φ HS ≈2
(4.58)